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http://dx.doi.org/10.4134/BKMS.b190624

GALKIN'S LOWER BOUND CONJECURE FOR LAGRANGIAN AND ORTHOGONAL GRASSMANNIANS  

Cheong, Daewoong (Department of Mathematics Chungbuk National University)
Han, Manwook (Department of Mathematics Chungbuk National University)
Publication Information
Bulletin of the Korean Mathematical Society / v.57, no.4, 2020 , pp. 933-943 More about this Journal
Abstract
Let M be a Fano manifold, and H🟉(M; ℂ) be the quantum cohomology ring of M with the quantum product 🟉. For 𝜎 ∈ H🟉(M; ℂ), denote by [𝜎] the quantum multiplication operator 𝜎🟉 on H🟉(M; ℂ). It was conjectured several years ago [7,8] and has been proved for many Fano manifolds [1,2,10,14], including our cases, that the operator [c1(M)] has a real valued eigenvalue 𝛿0 which is maximal among eigenvalues of [c1(M)]. Galkin's lower bound conjecture [6] states that for a Fano manifold M, 𝛿0 ≥ dim M + 1, and the equality holds if and only if M is the projective space ℙn. In this note, we show that Galkin's lower bound conjecture holds for Lagrangian and orthogonal Grassmannians, modulo some exceptions for the equality.
Keywords
Galkin's conjecture; property ${\mathcal{O}}$; Gamma conjectures;
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1 D. Cheong, Quantum multiplication operators for Lagrangian and orthogonal Grassmannians, J. Algebraic Combin. 45 (2017), no. 4, 1153-1171. https://doi.org/10.1007/s10801-017-0737-7   DOI
2 L. Evans, L. Schneider, R. Shifler, L. Short, and S. Warman, Galkin's lower conjecture holds for the Grassmannian, preprint.
3 W. Fulton, Young Tableaux, London Mathematical Society Student Texts, 35, Cambridge University Press, Cambridge, 1997.
4 W. Fulton and C. Woodward, On the quantum product of Schubert classes, J. Algebraic Geom. 13 (2004), no. 4, 641-661. https://doi.org/10.1090/S1056-3911-04-00365-0   DOI
5 S. Galkin, The conifold point, arXiv:1404.7388.
6 S. Galkin, V. Golyshev, and H. Iritani, Gamma classes and quantum cohomology of Fano manifolds: gamma conjectures, Duke Math. J. 165 (2016), no. 11, 2005-2077. https://doi.org/10.1215/00127094-3476593
7 S. Galkin and H. Iritani, Gamma conjectures and mirror symmetry, arXiv:1508.00719v2.
8 J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1972.
9 H. Ke, On Conjecture O for projective complete intersections, arXiv: 1809.10869.
10 D. Cheong and C. Li, On the conjecture O of GGI for G/P, Adv. Math. 306 (2017), 704-721. https://doi.org/10.1016/j.aim.2016.10.033   DOI
11 A. Kresch and H. Tamvakis, Quantum cohomology of the Lagrangian Grassmannian, J. Algebraic Geom. 12 (2003), no. 4, 777-810. https://doi.org/10.1090/S1056-3911-03-00347-3   DOI
12 A. Kresch and H. Tamvakis, Quantum cohomology of orthogonal Grassmannians, Compos. Math. 140 (2004), no. 2, 482-500. https://doi.org/10.1112/S0010437X03000204   DOI
13 A. Lascoux and P. Pragacz, Operator calculus for Q-polynomials and Schubert polynomials, Adv. Math. 140 (1998), no. 1, 1-43. https://doi.org/10.1006/aima.1998.1757   DOI
14 C. Li, L. C. Mihalcea, and R. M. Shifler, Conjecture O holds for the odd symplectic Grassmannian, Bull. Lond. Math. Soc. 51 (2019), no. 4, 705-714. https://doi.org/10.1112/blms.12268   DOI
15 P. Pragacz and J. Ratajski, Formulas for Lagrangian and orthogonal degeneracy loci; Q-polynomial approach, Compositio Mathematica 107; 11-87, 1997.   DOI
16 K. Rietsch, Quantum cohomology rings of Grassmannians and total positivity, Duke Math. J. 110 (2001), no. 3, 523-553. https://doi.org/10.1215/S0012-7094-01-11033-8   DOI